Bees Beat Machines At 'Traveling Salesman' Problem 394
eldavojohn writes "Recent research on bumble bees has proven that the tiny bee is better than computers at the traveling salesman problem. As bees visit flowers to collect nectar and pollen they discover other flowers en route in the wrong order. But they still manage to quickly learn and fly the optimally shortest path between flowers. Such a problem is NP-Hard and keeps our best machines thinking for days searching for a solution but researchers are quite interested how such a tiny insect can figure it out on the fly — especially given how important this problem is to networks and transportation. A testament to the power of even the smallest batch of neurons or simply evidence our algorithms need work?"
great... (Score:5, Funny)
Re:great... (Score:5, Funny)
Re:great... (Score:5, Informative)
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Huh, I didn't know that.
So if a "bee-wolf" upgrades the wolf to a bear, would a "bee-bear" (beobeowulf I guess?) turn it into a Tyranosaurus? Or a raptor with an RPG riding a shark?
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Either way, this problem sounds like it will keep computers buzzy.
Re:great... (Score:5, Funny)
Alas, it doesn't run Linux.
It run BEE-Os.
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Just imagine what we could have accomplished in computing if we'd stuck with B instead of moving on to C!
Re:great... (Score:4, Funny)
I always knew BeOS was underrated.
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Re:great... (Score:5, Funny)
Strangely enough, we also had a problem with a travelling salesman in my community, and we successfully used bees to deal with it:
http://www.youtube.com/watch?v=-1GadTfGFvU [youtube.com]
Bees have a guide (Score:2, Interesting)
After the genetic vector is put in, all the bees have to do is keep track of the sun. What amazes me though is how they look at another bee and visualize it traveling to a set patch of flowers, by looking at its dance.
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So, you're arguing it's the algorithm that's wrong and not a better set of neurons.
Re:Bees have a guide (Score:5, Informative)
What amazes me though is how they look at another bee and visualize it traveling to a set patch of flowers, by looking at its dance.
Are we discussing bumble bees or honey bees? The summary says bumble bees.
http://www.earthlife.net/insects/socbees.html [earthlife.net] states that bumble bees "...have not evolved any means of communicating information reguarding utilisable resources."
Wait, whut? (Score:3, Funny)
quite interested how such a tiny insect can figure it out on the fly
I thought we were talking about bees? I am so confused...
Re:Wait, whut? (Score:5, Funny)
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Looks like bees are the new buzzword.
Not for long... (Score:2)
Hah! They may be on top now, but thanks to CCD [wikipedia.org] we won't have to be #2 for long. Goooooooooo humans!
In Other ( Two ) Words: ( +1, Helpful ) (Score:5, Interesting)
Simulated annealing [wikipedia.org].
Yours In Akademgorodok,
Kilgore Trout.
Heuristic (Score:4, Insightful)
Is it possible that the honey bees aren't really solving the Traveling Salesmen problem at all, but rather employ some sort of unknown heuristic that leads to solutions that's close enough to optimal for it to look like that they've solved it? Maybe that's what we should be looking at rather than pondering if bees somehow have some sort of superior calculating ability over a supercomputer.
After all, when we're playing a game of baseball (right, right, I know, this is slashdot), and a ball is coming towards us, we aren't calculating in our heads the velocity, air resistance and other variables involved in catching the ball. We just reach out our arms and our brain makes its best guess based on some sort of heuristic or something to make the catch.
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your just playing word games... if i've developed a method of figuring out how to catch a ball without using newtonian physics, I've still solved the problem.
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Indeed - especially considering the fact that one of the major class of solutions based on simulated annealing is actually a heuristics based solution. It really doesnt matter how you solve it at all - any way is welcome - as long as it reaches close to the optimal solution.
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Nope. You've solved one small set of problems, which is related to the bigger problem. Does your method work for just balls? What about anvils? Churches? Very small rocks? How about when thrown through water, or over a hill? Can it be applied to every situation where Newtonian physics works?
The travelling salesman problem is a very specific definition of a problem. Solving it for one specific set (a given graph, size, or even geometry) does not solve the whole problem. Similarly, I know of no non-euclidean
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I've still solved the problem.
For yourself. However you are unable to break that problem down into tokens that can be taught to someone else or, say, programmed into a robot. Therefore although you might be good at catching a ball as an individual, you haven't solved the problem that those less adept than you face when put in the same situation.
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Unfortunately, when it comes to NP-complete, they world is not enough :-/
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Thanks for making the world a better place by taking your time to point out a typo. You are a man amongst men.
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Did you mean making the would a better place?
Different Spaces (Score:5, Insightful)
After all, when we're playing a game of baseball (right, right, I know, this is slashdot), and a ball is coming towards us, we aren't calculating in our heads the velocity, air resistance and other variables involved in catching the ball. We just reach out our arms and our brain makes its best guess based on some sort of heuristic or something to make the catch.
I think the problem with your analogy that there are an unlimited number of dimensions and responses where you could put your arm out to make the catch (well, not unlimited if you consider Planck distances to be the smallest possible distance). But when we are talking about computerized flowers with nectar, you pretty much can only go to one of the flowers next. I think they used RFID to track the bees (or at least this researcher has written about doing that before)? So we can sit there and do a star search on all paths of the 50 flowers and find the shortest one to connect all of them in three dimensions in a particular order (we assume the flight paths are straight lines). The difference is not that we have so many fewer things to search than in the ball catching example but that you take a very finite deterministic path (i.e. 2, 34, 23, 6, 18, etc) and the bees seem to be able to find and learn this very quickly. According to the researcher:
"In nature, bees have to link hundreds of flowers in a way that minimises travel distance, and then reliably find their way home - not a trivial feat if you have a brain the size of a pinhead! Indeed such travelling salesmen problems keep supercomputers busy for days. Studying how bee brains solve such challenging tasks might allow us to identify the minimal neural circuitry required for complex problem solving."
If this holds true for hundreds of flowers, I think we're talking about a serious search space with a definite path that is far more specific than the heuristics of moving your arm and hand around dynamically in space to collide with a ball. You could have tons of error when trying to catch a ball and still catch it. You (frequently) only have one optimal path in shortest distance problems. It's probably true these traveling salesman problems look obvious to a bee like catching a ball does to us but something particularly interesting is going on there if it is.
Let's say it is an unknown heuristic. I'd wager the network folks would kill to know how that heuristic is so cheaply computed.
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Good point, but what I meant by the baseball analogy isn't about the unrestrictedness or complexity of the problem so much as simply an example of a heuristic that we naturally have (although dependent on our personal levels of coordination). We're descended from tree-living ancestors who naturally developed the ability to judge an object's distances and movement (otherwise, our ancestors would've just fallen out of trees or failed to grab a branch as it jumps). Likewise, a mountain goat will be able to nat
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I'd be interested to read the full paper... looks like it won't be published until this week.
I have to wonder if it's simply local optimization that the bees are using - i.e., "Fly to a close flower not yet visited" - that starts looking like they're solving a much more complex problem? Are they visiting *every* flower on the "map"? Are they ever skipping some? Do they visit the flowers in exactly the same order, or is there variance from bee to bee (or between two trips from the same bee?)
It seems to me
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That's true, although you could still reduce it back to the TSP easily enough - optimize for 'minimal energy expenditure' instead of 'minimal distance,' since distance traveled factors into the energy expended. Of course, even that still varies in 'real world' conditions - breezes come and go and change direction, birds come by looking for a snack, a lawnmower destroys a doz
Not the TSP (Score:5, Interesting)
Is it possible that the honey bees aren't really solving the Traveling Salesmen problem at all, but rather employ some sort of unknown heuristic that leads to solutions that's close enough to optimal for it to look like that they've solved it?
This article is fundamentally misstating the TSP. If it were the TSP, the bees wouldn't get to choose their route.
As other bees come in and report their route taken and pollen collected, another bee will put bits of those routes together. (Which would be the surprisingly difficult part to me, since the bees are doing some pretty complicated vector algebra.) But a bee is going to have a budget of so much daylight and will attempt to maximize the amount of nectar it collects in that time, given the bits of routes collected by other bees and its own scouting. But it's not given a list of points it has to hit, it picks its list from a larger list of points. That's fundamentally different from the TSP, even solving it by heuristic.
Re:Not the TSP (Score:4, Insightful)
The questions this raises are:
1) Do bees always visit every flower (node) on the map?
2) Once they've calculated their "optimal" route, do they ever vary it?
3) If it's a heuristic - does it scale? Will it work for more than 100 flowers spread across something the size of my backyard? Or is the heuristic going to break down or become completely unworkable once the number of nodes reaches a certain point?
What we can say so far is that they "appear" to be solving the problem, for some limited subset of the problem space. In actuality, they may be using some very simple rules that approximate the solution for small numbers of nodes and distances, but which would result in inefficient or sub-optimal solutions on a larger scale.
In other words - I can catch a pop-fly to right field. Does that mean I can also use the same heuristics I'd use to catch a small object at low speeds over small distances to accurately launch a Patriot Missile to intercept an ICBM? The physics are the same, but a little jitter of a couple millimeters in my calculations when applied to a distance of several hundreds or thousands of kilometers will result in a pretty big miss.
Re:Heuristic (Score:5, Interesting)
After all, when we're playing a game of baseball (right, right, I know, this is slashdot), and a ball is coming towards us, we aren't calculating in our heads the velocity, air resistance and other variables involved in catching the ball. We just reach out our arms and our brain makes its best guess based on some sort of heuristic or something to make the catch.
You should read "On Intelligence" if you're at all interested in that subject. Jeff Hawkins (Palm inventor) proposes a fascinating theory of the inner workings of the brain.
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Thanks, man, I'll check it out.
Re:Heuristic (Score:4, Informative)
That's pretty much what I was going to post. The bees almost certainly aren't solving the Traveling Salesman problem, they're getting good enough approximations of a solution. Our computers don't chug for days trying to figure out the answer to TSPs, they chug for a couple of seconds and produce a close-to-optimal solution.
And the thing is, not all instances of the TSP are necessarily NP-Hard (for instance: if there was only one road between each city + 1 extra road between the first and last cities, the optimal solution is obvious), and the cases of it found in practical applications are generally far easier to handle than the cases found in more esoteric theoretical constructs (for instance: if you move east, you move closer to all the flowers in the east; this is not necessarily true in the general TSP). Most real instances of the TSP can be handled well enough with simple, quick greedy algorithms; they won't necessarily give you the best answer, but it'll be pretty close.
It don't matter what you call it (Score:2)
The travelling salesman problem is the problem of finding the shortest route between a set of points. It doesn't matter HOW you solve it. You could time all possible journey's, you could do a sorting routine or god knows what. But if you solve it, you solved it.
That is what the bee does. And maybe if we can learn HOW the bee does that, we might learn something from it. It might be a smarter way of solving things. Or maybe Bees have an additional variable from an unknown input that helps solve it.
And as fo
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The travelling salesman problem is the problem of finding the shortest route between a set of points. It doesn't matter HOW you solve it. You could time all possible journey's, you could do a sorting routine or god knows what. But if you solve it, you solved it.
That is what the bee does. And maybe if we can learn HOW the bee does that, we might learn something from it. It might be a smarter way of solving things. Or maybe Bees have an additional variable from an unknown input that helps solve it.
That's not what the bee does. That is what someone _claims_ the bee does. Big, big difference. I imagine they found that the bees visit flowers faster than in random order, or faster than the order in which they were told about the flowers. I think a simple algorithm like "always visit the nearest flower that wasn't visited yet" would probably impress these "scientists" no end.
BTW. Bees shouldn't even try to solve the travelling salesman problem. Assume a single flower very far away - the best solution m
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[...] our brain makes its best guess based on some sort of heuristic or something to make the catch.
Maybe your brain, but not mine. Mine generally makes me miss the ball by 3 feet or get hit in the nads. Maybe my brain works like old Intel processors?
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I beg to differ. If it takes a supercomputer 4 days to solve it for 5 flowers... we have huge problems.
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Your tax dollars at work...
Re:Heuristic (Score:5, Funny)
It's called Ruby, and it's not a problem, it's a feature, a trendy one.
I doubt it (Score:3, Insightful)
First and foremost, how many nodes are we talking about here? I highly doubt that the bees are keeping track of hundreds of feeding spots from one trip out to the next but the article doesn't say.
The second problem is this "Computers solve it by comparing the length of all possible routes and choosing the shortest." Who on earth would try to solve the traveling salesman this way? Yeah, a brute force solution will get you the guaranteed best path, but the performance is horrible. There's lots and lots of shortcuts that can save a huge amount of time, things as simple as eliminating crossed paths can make a big difference. You can even use techniques like genetic engineering successfully on such a problem (though you might not reach the absolute best path that way).
Re:I doubt it (Score:5, Insightful)
I think you mean genetic algorithms. Genetic engineering is...something else.
Re:I doubt it (Score:5, Funny)
(I assuming we can engineer Hulks.)
The answer is obvious (Score:5, Funny)
Shortcuts (Score:5, Interesting)
Re:Shortcuts (Score:5, Insightful)
Which makes the problem more difficult, not less. The way it is usually presented in CS the distance between the nodes is the minimum cost path, the bees would also have to 'calculate' that in addition to solving for the correct order. Think about it this way, imagine trying to solve the traveling salesman path between 100 cities, but you can take any route you want between cities. You could take all the back roads, the freeway, you could hop on a train or an airplane, you could kayak down the river between two cities. It doesn't make the problem any easier, in fact it adds a ton more variables to the mix, effectively increasing the number of routes that would need to be checked using a brute force solution.
Re:Shortcuts (Score:5, Funny)
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Re:Shortcuts (Score:5, Funny)
I dunno. If only we had a word for this....something like the line that a bee would travel...
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Well... (Score:5, Funny)
Does this mean that B >= NP?
Entomoengineering? (Score:5, Interesting)
We have a lot yet to learn from our six-legged colleagues, from the sound of it. Recently some work was done on optimizing machine vision using an algorithm derived from the way the house fly's vision works. [uwyo.edu] The termite's wood-digesting gut is a prime object of study for those seeking to manufacture fuel from biomass efficiently and cleanly. An insect virus (the baculovirus) is the new hotness for gene transduction in mammalian cells because it can't actually cause disease.
I think this might be the next step in bioengineering. We've been grabbing genes out of various organisms and sticking them in bacteria to produce useful biomolecules like insulin and factor VIII. Maybe the insect is our next stop.
Re:Entomoengineering? (Score:5, Funny)
Re:Entomoengineering? (Score:4, Funny)
Oh, really? (Score:4, Interesting)
But really no details are given. Do the bees still travel to all of the flowers? I'd imagine they might just decide to skip one or two if they don't fall close enough to the path to be worth it. They don't say what they did (probably nothing) to validate that the bees actually found the shortest path. Did the "graph" that they gave the bees include a section where a greedy algorithm would fail? What is more likely is the bees haven't solved it, but found a decent approximation.
I think this is what you get when you let bee researchers write math/computer science articles.
Re:Oh, really? (Score:5, Insightful)
100 flowers=100! possibilities. Using brute force on a 1 GHz processor and computing one solution per cycle (quite optimistic), it would take you 3 times 10 to the 141 years to complete. Even if your cellphone had a helaflop [slashdot.org] processor, it would still take longer than the age of the universe to compute this way.
hierarchical models (Score:2, Informative)
Wild Guess (Score:2)
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Like ants with wings!
Of course, the ant solution still isn't very fast, or reliable, and is usually far slower than algorithmic solutions. Nice try, though.
Re:Wild Guess (Score:4, Interesting)
Nearest Neighbor? (Score:2)
I read TFA and it seems more focused on the excitement that the bees can solve the TSP, but the researchers never seem to indicate how the bees are doing it, and given the nature of the problem, how do they know it really is the "optimum" solution. Based on my limited work with the TSP, the only algorithm that, for my purposes, has worked the best is Nearest Neighbor, which is also, I believe, the simplest but also most naive.
Would be interesting to know what the bees' algorithm is.
Grass seed? You mean CORN? (Score:5, Funny)
Gosh, that is one hell of a bee if it has the brain of a piece of corn... or is corn not a grass anymore? At least when you take some idiotic comparison, take one that has a non-changing size. Penny is okay because all pennies at least within a country tend to stay the same. But grass seeds?
Next up is "brain the size of a pinhead". Oh okay, so there are many sizes of pin but at least we can assume some kind of standard. And that is FAR smaller then most grasses I know and see seeds of in Holland.
Otherwise intresting stuff but I loathe this "make it easier" by obuscating the facts.
Number of neurons in honey bee brain = 950,000 (from Menzel, R. and Giurfa, M., Cognitive architecture of a mini-brain: the honeybee, Trd. Cog. Sci., 5:62-71, 2001.)
Now THAT is a fact. We? We got 100 billion. So, while a bee has a tiny brain compared to ours, it is HARDLY simple. And because it is far smaller and far more primitive it doesn't need as as much intelligence to deal with things it doesn't need to. Listening and producing speech is complex, but bees don't bother with that. Living for half a century and remembring everything is complex. But bees don't do that.
This why computers can do math so fast despite being so stupid, because they only do math.
How can the bee do route calculation with close to a million neurons? I have no idea but didn't research show that far fewer rat neurons could fly a plane? I think some people fastly underestimate the complexity of the brain even small ones. We already know that a neuron is far more then a simple transistor so 1 million super transistors would make for a hell of a complex computer. Suddenly it doesn't seem to odd that a bee can do computations far more complex because THAT is what it is designed to do. You could just as easily marvle at the fact that the bee with its tiny brain can fly, while I with my large brain can't. And no I don't just mean I don't have wings, I mean that if you put me in a helicopter you would have a horrible crash in seconds and that is in the passenger seat.
Marvle at nature, learn from it but don't belittle it. It takes us year to program a robot to walk very very slowly. A deer learns it in minutes and this includes learning to control legs locked up in a womb for months. We can either accept that nature is amazing or we are very very poor programmers... as a developer, I choose to believe that nature is amazing.
The computer isnt going to die (Score:3, Insightful)
The computer isn't going to die if it doesn't get the right path, the bee might. Death is a remarkably strong motivator to be efficient.
Re:The computer isnt going to die (Score:5, Funny)
The computer isn't going to die if it doesn't get the right path, the bee might. Death is a remarkably strong motivator to be efficient.
Don't tell my boss.
Natural selection at work (Score:2)
Any colony optimisation (Score:2)
Or just use Ant Colony Optimisation, which has been doing this for 18 years.
Not really news (Score:2)
165 million years of evolution, anyone? (Score:2)
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No I for one? (Score:3, Funny)
No "I for one welcome our new insect overlords"? Who are you and what have you done with Slashdot?
Slime molds do something similar (Score:5, Interesting)
Bullshit (Score:3, Insightful)
So they have proof that these bees solve the travelling salesman problem? Not just get a good approximation? Not just solve a slightly constrained version? Not just solve a slightly different problem? You know all the things that computers do just fine thank you very much, and aren't NP-hard.
I notice the journal is a Naturalist one and the researchers aren't are bioligists and chemists not computer scientists.
I have no difficulty believing bees have evolved (or been designed with if you must for those I don't feel like arguing with) a very efficient way of collecting pollen - it is after all fundamnental to their survival and reproduction. But that they happened to solve an NP-Hard problem that they have no need of solving (does an individual bee really visit *every* location on one trip? surely some imperfection would help in discovering new plants by having bees follow different paths?) - that seems a bit of a stretch.
Bogus claim (Score:5, Informative)
Oh, this one again. I've seen this claim made for neural nets back in the 1980s, and for DNA computers in the 2000s. It's bogus.
The guaranteed-optimum solution to the TSP is NP-hard. The "gets to the optimum 99% of the time and close to it all the time" solution is easy. It was developed at Bell Labs in the 1960s. Here it is:
This is a particularly efficient way to do it. I once coded this for a PC/AT, and it took less than a second for N=50. Almost any scheme which randomly breaks links and tries to improve the path will eventually converge on a near-optimum solution.
Massively Multithreaded Genetic Algorithm (Score:3, Interesting)
I'd be willing to bet it has something to do with the fact that you have an entire hive of bees each attempting to find the shortest path and then sharing their experience via that 'bee dance' thing (http://www.youtube.com/watch?v=-7ijI-g4jHg). Each bee is a thread with its own particular solution to the problem. Each bee's behavior contributes random heuristic alterations to the nectar-gathering path based on bee instincts evolved over millions of years. The bees periodically exchange solutions via the bee dance. It's a classic Genetic Algorithm.
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who/what is god?
Re:Evidence (Score:5, Funny)
It's an abbreviation for Good Old Darwinism.
Re:Evidence (Score:4, Funny)
Thereby substituting one guy with a beard who must be worshipped for another.
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who/what is god?
I think it's one of the security modes in Linux ... I know one of the user modes in Windows is called "damn!".
Re:Evidence (Score:4, Funny)
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What Slashdot needs is a write-in moderation option so you can moderate (-1, Wrong) or (+1, Fucking Awesome).
Solving a different problem (Score:5, Informative)
The canonical traveling salesman problem usually is states that all cities must be visited. The bee is not under this constraint. This changes the problem from a do-or-fail NP hard problem to a more simple approximate optimization problem. The latter have many many many many many good solution paths in computers. Perhaps the newest and best approach that resembles the bee's agent based learning approach is called Probability Collectives (google it). You'll want to learn it since it works well on parallel computers, distributed computing, and most of all on systems composed on many dumb subunits on a sparsely connected network with no central command and control (think mobile devices).
Re:Solving a different problem (Score:5, Insightful)
I suppose the exception is when competing against an intelligent adversary, who constantly strives to give you worst-case problems and where a small margin of victory is a victory nonetheless.
Probability Collectives (Score:3, Informative)
Probability Collectives are interesting because they are one of very few optimization alogorithms born from first principles considerations. For example, Simmulated annealing comes from Metropolis/hastings search and that was a brilliant breakthrough that allowed rapid exploration in a way that guarentees detailed balance. Parallel Tempering is the parallel extension of that first-principles argument. Most other search and optimization algorithms are born from either heurisitics expected to align with
Re:Solving a different problem (Score:4, Interesting)
I believe the bee's have an advantage over the typical traveling salesman problem in that the bee's are finding the shortest path on a fully connected or complete graph. The traveling salesman problem is hard because the graph is not necessarily fully connected so all paths have to be examined individually. The bee presumably also has a predetermined starting node, the one closes to where it is released.
I believe the shortest path on a fully connected graph is found by always choosing the closest non-visited neighbor from the current node. The difference in calculation is O(n!) vs. O(n^2).
Re:Evidence (Score:5, Insightful)
You mean millions of iterations of random chance have selected the most efficient pollen-gatherers.
Re:Evidence (Score:4, Insightful)
That's a simple way of saying that evolution created a program in their brains that solves the TSP fairly efficiently under certain constraints.
Re:Evidence (Score:5, Insightful)
Anyone who thinks the bees solved NP-hard problem here does not know what they're talking about...
Those bees did not do an exhaustive search for the optimal path, only one that is 'good enough'. Computers can do the same with any decent algorithm. Only difference here i assume is that the bees have hardware that has gone through natural selection to produce a pretty good 'best effort' algorithm.
If those bees can produce the optimal path for all corner-case setups then I'll be rather impressed.
Re:Evidence (Score:4, Interesting)
Those bees did not do an exhaustive search for the optimal path, only one that is 'good enough'.
How do you know this? I'm not seeing that stated in the article. In fact,
Scientists at Queen Mary, University of London and Royal Holloway, University of London have discovered that bees learn to fly the shortest possible route between flowers even if they discover the flowers in a different order.
This seems to directly contradict what you're saying, so I'll assume you have access to more information and will be linking likewise shortly...
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Thanks - this is what I was thinking too. It seems like the hard part of TSP is to *prove* that the shortest solution is in fact shortest. I think it's relatively trivial to find a very short solution to the problem (even by hand for a reasonable # of nodes). Demonstrating that the proposed path is in fact the absolute best is where the headaches (NP time) come in?
Re:Evidence (Score:5, Insightful)
Question is...did the bees evolve to find the corner cases, or did the plants evolve so the damn bees could find them in the first place? After all, plants that are stupid enough to hide from bees while simultaneously needing them to reproduce would stand a good chance of not making it to the next generation ;-)
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Sure, chemistry no one has been able to reproduce in the lab under any circumstances.
No, it's based on basic biochemical reactions which are demonstrated on a daily basis in any number of labs. Are you completely ignorant of that field, and simply repeating dogmatic statements with no correlation to reality?
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Actually I worked this problem years ago, despite having not found a solution, I was able to determine the problem is fundamentally far simpler than traveling salesman as the nodes are distributed on a sheet with simple calculations.
So what is the solution? Do you sleep with the farmer's daughter or sleep in the barn?
Maybe I'm thinking of a different "traveling salesman" problem.
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So what is the solution? Do you sleep with the farmer's daughter or sleep in the barn?
Why choose? Haven't you heard of a "roll in the hay" before?
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So long as it's not the Crushinator, you ALWAYS sleep with the farmer's daughter.
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Also, there is a constrained version of the problem called Bitonic Tours [wikipedia.org] which is solvable in poly time, which might match the flower scenario well.